Results like above rarely holds in higher dimensions, for instance, Rudin showed that even in \(H^2(2)\)  such a result does not exist (see [1]}). However, in the case of unit ball there are several rkHs's which admit Beurling-type characterization of invariant subspaces. A recent result by Clouâtre et al. in [2]} proved that rkHs's with cnp factors admit this property. The list of references in [2]} gives a brief account of several earlier results in this direction. For our interest, we combine [Theorem 1.1, Proposition 2.5, [2]}] to obtain the following result.
